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Chimeras in random non-complete networks of phase oscillators
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10.1063/1.3694118
/content/aip/journal/chaos/22/1/10.1063/1.3694118
http://aip.metastore.ingenta.com/content/aip/journal/chaos/22/1/10.1063/1.3694118
View: Figures

Figures

Image of FIG. 1.
FIG. 1.

(Color online) A chimera solution of Eqs. (1) and (2) for full connectivity matrices. Top: a temporal snapshot of the phase as a function of index, i, where those in population 1 are numbered 1 to 500, and those in population 2 are numbered 501 to 1000. Bottom left: histogram of oscillator phases in population 1. Bottom right: histogram of oscillator phases in population 2. Parameter values: . All chosen from a Lorentzian distribution with half-width-at-half-maximum 0.001.

Image of FIG. 2.
FIG. 2.

(Color online) Steady states of Eqs. (29) and (30) , where . This figure relates to the all-to-all coupled case in the limit . Lines indicate symmetric solutions for which ; solid blue: stable, dashed red: unstable. Points indicate chimera states where ; blue: stable, red: unstable. The symmetric state undergoes two pitchfork bifurcations as is varied. The lower panel is an enlargement of the upper panel. Parameters: .

Image of FIG. 3.
FIG. 3.

(Color online) Bifurcations of fixed points of Eqs. (29) and (30) . Green dots: Hopf bifurcation of the stable chimera; blue line: saddle-node bifurcation of the stable and unstable chimera; red dotted: pitchfork bifurcation of the symmetric state in which . A stable chimera exists in the region bounded by the left-most pitchfork bifurcation, the saddle-node bifurcation and the Hopf bifurcation. Figure 2 is a horizontal “slice” through this figure at E = 0.2. Parameter: .

Image of FIG. 4.
FIG. 4.

(Color online) Hopf (dashed) and saddle-node and left-most pitchfork (solid) bifurcations of fixed points of Eqs. (29) and (30) for (black, also shown in Fig. 3 ), (blue) and (red). The arrows indicate how the curves move as is increased. A stable chimera exists within the region bounded by curves of the same colour, for the corresponding value of .

Image of FIG. 5.
FIG. 5.

(Color online) Degree distributions for Erdös-Rényi-type networks (top row) and Chung-Lu-type networks (bottom row). See text for explanation of parameters. Here N = 300.

Image of FIG. 6.
FIG. 6.

(Color online) A chimera solution in a network with Erdös-Rényi-type connectivity. Oscillators in population 1 are numbered 1 to 300, those in population 2 are numbered 301 to 600. Top and middle panels show fixed points of Eqs. (31) and (32) , while the bottom panel is a snapshot of Eqs. (1) and (2) for the same connectivity matrices. Top: magnitudes of the (left) and (right). Middle: argument of the (left) and (right). Other parameters: .

Image of FIG. 7.
FIG. 7.

The values of for which there is a saddle-node bifurcation of fixed points of Eqs. (25), (26), (27), (28), and (31) and (32) for Erdös-Rényi-type random matrices A, B, C, and D, as a function of p. At each value of p, 10 realisations of the random matrices were used, and the mean and standard deviation of the 10 are shown. Other parameters: .

Image of FIG. 8.
FIG. 8.

Values of E at which a Hopf bifurcation of a stable stationary chimera state occurs for an Erdös-Rényi-type network, for different values of p. At each value of p, 10 realisations of the random matrices were used, and the mean and standard deviation of the 10 values of E are shown. Other parameters: .

Image of FIG. 9.
FIG. 9.

(Color online) Onset of a Hopf bifurcation caused by randomly removing (with equal probability) a few of the connections within the phase oscillator network (1) and (2) . At t = 1000 we switched the connectivity matrices A, B, C, and D from full to Erdös-Rényi-type matrices with p = 0.9. We define and . Other parameters: .

Image of FIG. 10.
FIG. 10.

The values of for which there is a saddle-node bifurcation of fixed points of Eqs. (25), (26), (27), (28), and (31) , (32) for Chung-Lu-type random matrices A, B, C, and D, as a function of r. At each value of r, 10 realisations of the random matrices were used, and the mean and standard deviation of the 10 are shown. Other parameters: .

Image of FIG. 11.
FIG. 11.

Values of E at which a Hopf bifurcation of a stable stationary chimera state occurs for a Chung-Lu-type network, for different values of r. At each value of r, 10 realisations of the random matrices were used, and the mean and standard deviation of the 10 values of E are shown. Other parameters: .

Image of FIG. 12.
FIG. 12.

(Color online) Suppression of oscillations by removing some of the connections within the oscillator network (1) and (2) . At t = 1000 we switched the connectivity matrices A, B, C, and D from full to Chung-Lu-type matrices with r = 0.1. We define and . Other parameters: .

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/content/aip/journal/chaos/22/1/10.1063/1.3694118
2012-03-19
2014-04-23
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Chimeras in random non-complete networks of phase oscillators
http://aip.metastore.ingenta.com/content/aip/journal/chaos/22/1/10.1063/1.3694118
10.1063/1.3694118
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